Imaginary number

An imaginary number is the square root of a negative real number. (The
square root of a number is a second number that, when multiplied by
itself, equals the first number.) As an example, √−25 is an
imaginary number.

The problem with imaginary numbers arises because the square (the result
of a number multiplied by itself) of any real number is always a positive
number. For example, the square of 5 is 25. But the square of
−5 (−5 × −5) is also 25. What does it mean,
then, to say that the square of some number is −25. In other words,
what is the answer to the problem √−25 = ?

As early as the sixteenth century, mathematicians were puzzled by this
question. Italian mathematician Girolamo Cardano (1501–1576) is
generally regarded as the first person to have studied imaginary numbers.
Eventually, a custom developed for using the lowercase letter
i
to represent the square root of a negative number. Thus √−1
=
i
, and √−25 = √25 × √−1 = 5
i
.

Complex numbers

Imaginary numbers were largely a stepchild in mathematics until the
nineteenth century. Then, they were incorporated into another mathematical
concept known as complex numbers. A complex number is a number that
consists of a real part and an imaginary part. For example, the number 5 +
3
i
is a complex number because it contains a real number (5) and an
imaginary number (3
i
). One reason complex numbers are important is that they can be
manipulated in ways so as to eliminate the imaginary part.